## Analysis of Time Series

Table of Contents

### Q.2) Write a short essay on “analysis of time series”.

**Answer:****ANALYSIS OF TIME SERIES**

A time series analysis is mainly concerned with the decomposition of observed series. Data into its component so as to estimate their separate effects to do this, we must make separate assumption about the relationship existing among the various components. According, theses component are assumed to have either the multiplication relationship or additive relationship or model. The analysis of time series data is of particular interest to the businessman because a large proportion of basic data with which he must deal are time series data.

**Q.3) What is a time series? What are its main components? Describe each briefly.**

**Answer:**

When the data is arranged at more or less regular interval of time is called a time series. In other words a time series consists of numerical data collected, observed or recorded at more or less regular intervals of time each hour, day, quarter or year. More specifically, it is any set of data in which observations are arranged in a sequential order.

The main components are secular trend (T), seasonal variation (S), cyclical fluctuations (C) and irregular or random variation (I).**Secular trend:-**

A secular trend is a long term variation that persists for many years and indicates the general direction of the change of observed values. In other words, it refers to a smooth, broad movement of a time series in the same direction, showing a gradual rise or fall within the data.**Seasonal variations:-**

The seasonal variations, which are namely caused by the change in seasons, are short-term movements occurring in periodic manner. These fluctuations are repeated with more or less the same power within a specified period of one year or shorter. The main causes for seasonal variations are the weather conditions, the religious festivals and the social customs. Examples of seasonal variations are the prices of wheat, which fall after the harvesting season and rise before the sowing time, the sales of soft drinks, which are high in the summer and low in the winter.**Cyclical fluctuations.-**

The cyclical fluctuations are the long period oscillations about the long-term trend, which tends to occur in a or less regular pattern over a period of certain number of years. The so-called business cycles which represent alternating period of richness and sorrow, provide an important example of cyclical movements. A cycle is said to be completed when beginning with a peak, the decline curve reaches a low point, called a trough and then rising again reaches the next peak.**Irregular or random variations:-**

These variations are irregular and unsystematic in nature. They occur in a completely unreliable manner as some unusual events such as floods, strikes, wars etc. these variations are also known as the accidental variations.

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**Q.4) Describe the different components of time series. Give advantages and disadvantages of any one of them.**

**Answer:****Secular trend:-**

A secular trend is a long-term variation that persists for many years and indicates the general direction of the change of observed values. In other words, it refers to a smooth, broad movement of a time series in the same direction, showing a gradual rise or fall within the data.**Seasonal variations:-**

The seasonal variations, which are namely caused by the change in seasons, are short-term movements occurring in periodic manner. These fluctuations are repeated with more or less the same power within a specified period of one year or shorter. The main causes for seasonal variations are the weather conditions, the religious festivals and the social customs. Examples of seasonal variations are the prices of wheat, which fall after the harvesting season and rise before the sowing time, the sales of soft drinks, which are high in the summer and low in the winter.**Cyclical fluctuations.-**

The cyclical fluctuations are the long period oscillations about the long-term trend, which tends to occur in a more or less regular pattern over a period of certain number of years. The so-called business cycles which represent alternating period of richness and sorrow, provide an important example of cyclical movements. A cycle is said to be completed when beginning with a peak, the decline curve reaches a low point, called a trough and then rising again reaches the next peak.**Irregular or random variations:-**

These variations are irregular and unsystematic in nature. They occur in a completely unreliable manner as some unusual events such as floods, strikes, wars etc. these variations are also known as the accidental variations.**Advantages and disadvantages of seasonal variations:-**

The study and measurement of seasonal variation is very important in certain cases. In order any store successfully, it is necessary to study the seasonal variation in the demand of various items very carefully. For example the sale of warm clothes increases in winter and decreases in summer.

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**Q.5) What is meant by secular trend? Describe the various methods of measuring a secular trend in a time series.**

**Answer:****Secular trend:-**

A secular trend is a long-term variation that persists for many years and indicates the general direction of the change of observed values. In other words, it refers to a smooth, broad movement of a time series in the same direction, showing a gradual rise or fall within the data.**Methods for the measurement of secular trend are:-**

· The method of freehand curve.

· The method of semi averages

· The method of moving averages

· The method of least squares.

**The method of freehand curve.**

Plot the given data on a graph paper and join the plotted points by segment of straight line. Observe the up and down movements on the graph and draw a smooth curve or a straight-line freehand passing through the plotted points in a way such that the general direction of change in values is indicated.**The method of semi averages.**

Divide the values in the series into two equal parts, including the middle value in each half or omitting it altogether when the number of values is odd. Find the average of the values in each part and place the average value against the respective midpoints of the two parts. Plot these two average values on the graph of the original values, draw a straight line connecting the two points and extend the line to cover the whole series. This is the semi average trend line.**The method of moving averages.**

The moving averages of period k of a time series are averages, each of k successive observations of the time series. We start with the first k observations, then leaving the 1st and including the (k+Dth observation, and repeating the process until we arrive at the last k observation. Each average is placed against the midpoint of the time interval covered in the calculation of the moving average.**The method of least square.**

A trend line, which can be described by a mathematical equation of the form of a straight line, a parabola can be measured using the method of least squares, by letting time to be the independent variable. Let us suppose that the long-term trend appears to be linear. Then the equation of the linear trend would be Y = a + b X the values of a and b, the two constants, are determined by solving the two normal equations simultaneously. The trend values are computed from this equation by substituting the values of X and are plotted on the graph of the original values.

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**Q.6) Discuss the advantages and disadvantages of any three methods of smoothing the data.**

**Answer:****Advantage of free hand curve method.**

- The method is very simple and if drawn carefully, the trend fitted by this method will be close approximation to a mathematically fitted trend.
- This method is quite flexible, can be used for all types of trend, linear or non-linear and requires a minimum of labor.

**Disadvantages.**

- This method is subjective, because different people will draw the free hand curves with different slopes leading to different conclusion.
- The method assumes no mathematical law of the trend, it is not suitable for forecasting purposes.

**Advantages of semi average method.**

- The methods is very simple and quick
- It does not depend on personal judgment

**Disadvantage**

- This method makes use of the arithmetic mean, which is greatly affected by extreme values.
- This method is not suitable when the trends is’ non linear

**Advantage of moving average methods**

- The methods of moving averages is easy and simple ,the moving averages of appropriate period estimate the combined effects of trend and cyclical components and give the smooth version of series by removing the seasonal and other effects .

**Disadvantages**

- The moving average trend does not provide values at the end or the beginning of the original series.
- This method does not give the trend for the whole duration.
- This method assumes no law of change but it cannot be used for forecasting purpose

**Advantages of least square method**

- This method is mathematically sound and therefore useful for prediction purposes.
- It is also an objective method

**Disadvantages**

- It is some time difficult to choose a proper mathematical equation
- In the case of nonlinear trends, the method may involve heavy calculations.

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